Abstract
We show that Bochner-flat para-Kähler surfaces are self-dual Walker manifolds and therefore they are locally isometric to the cotangent bundle of an affine surface equipped with a modified Riemannian extension. Explicit examples of constant and non-constant scalar curvature are given.
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1 Introduction
A para-Kähler manifold is a symplectic manifold \((M^{2n},\Omega )\) that is locally diffeomorphic to a product of Lagrangian submanifolds. This way its tangent bundle decomposes as a Whitney sum of Lagrangian subbundles \(TM = L\oplus L'\). Considering \(\pi _L\) and \(\pi _{L'}\) the projections on each subbundle, the (1, 1)-tensor field defined by \(J=\pi _L-\pi _{L'}\) is an almost paracomplex structure on M. Moreover, since L and \(L'\) are Lagrangian subspaces one has that \(\Omega (JX,JY) = -\Omega (X,Y)\) for all vector fields X, Y on M and so \(g(X,Y) = \Omega (JX,Y)\) defines a neutral signature metric on M such that \(g(JX,JY) = -g(X,Y)\) and \(\nabla J=0\), where \(\nabla \) denotes the Levi-Civita connection of (M, g).
Para-Kähler structures, which are also called bi-Lagrangian manifolds in the literature, are relevant for both Physics and Geometry. Para-Kähler geometry plays an important role in the study of several geometric problems such as the non-uniqueness of the metric for the Levi-Civita connection [5], the classification of symplectic connections [7], the spaces of oriented geodesics [3], the study of cones over pseudo-Riemannian manifolds [2] or the classical Monge-Kantorovich mass transport [15] (see also [11] for applications to supersymmetry). Paracomplex geometry is also relevant for understanding Weierstrass and Enneper type representations for Lorentzian surfaces in \(\mathbb {R}^{2,1}\) [10, 16].
The Bochner curvature tensor was introduced by S. Bochner in 1949 [4]. It is formally defined as an analogue of the Weyl curvature tensor, so that the curvature tensor of a Bochner-flat manifold is completely determined by its Ricci tensor. Let \((M^{2n},g,J)\) be a para-Kähler manifold. Its Bochner curvature tensor is defined as
for all vector fields X, Y, Z on M, where
A para-Kähler manifold is said to be Bochner-flat if its Bochner tensor vanishes identically. A para-Kähler manifold has constant paraholomorphic sectional curvature c if and only if its curvature tensor is of the form \(R(X,Y)Z = \frac{c}{4}R_0(X,Y)Z\) (see [14]). This way, any para-Kähler manifold of constant paraholomorphic sectional curvature is Bochner-flat. Moreover, a Bochner-flat para-Kähler manifold has constant paraholomorphic sectional curvature if and only if it is Einstein.
Even though the condition of being Bochner-flat is somehow analogous to that of being locally conformally flat, it is more restrictive since a Bochner-flat para-Kähler manifold has constant scalar curvature if and only if it is locally symmetric [17]. Moreover, if its Ricci operator is diagonalizable then the manifold either has constant paraholomorphic sectional curvature or it is locally isometric to a product of two spaces of constant opposite paraholomorphic sectional curvature.
The anti-self-dual Weyl curvature tensor of a four-dimensional para-Kähler manifold is determined by its scalar curvature as \(W^- = \frac{\tau }{12}{\text {diag}}[2,-1,-1]\) and the symplectic form \(\Omega \) is an eigenvector for the distinguished eigenvalue. On the other hand, the self-dual Weyl curvature tensor of a para-Kähler manifold is completely determined by the Bochner tensor, so \(W^+ = 0\) if and only if the manifold is Bochner-flat (see [6]). An immediate consequence of these facts is that a four-dimensional para-Kähler manifold is locally conformally flat if and only if it is Bochner-flat and its scalar curvature vanishes identically.
Let (M, g, J) be a para-Kähler manifold and denote \(\mathfrak {D}_{\pm } = {\text {ker}}(J\mp {\text {Id}})\) the eigenspaces corresponding to the eigenvalues \(\pm 1\) of the paracomplex structure J. \(\mathfrak {D}_{\pm }\) are parallel degenerate distributions and so any para-Kähler surface has an underlying Walker structure. This fact allows us to study para-Kähler structures through Walker manifolds.
The present work is organized as follows. Section 2 is devoted to the description of Walker structures in dimension four, paying special attention to self-dual Walker structures, in order to pave the way for the understanding of Bochner-flat para-Kähler structures in Sect. 3. Note that the para-Kähler and the Walker structures induce distinguished opposite orientations on the manifold, a fact that plays an important role in the theory. The classification of Bochner-flat para-Kähler surfaces of constant scalar curvature is given in Theorem 2, specifying the different curvature models realized in each situation. Finally, some examples of Bochner-flat para-Kähler surfaces of non-constant scalar curvature are provided in Sect. 3.2.
2 Walker Structures
Let \((M,g,\mathfrak {D})\) be a four-dimensional Walker manifold, i.e. a pseudo-Riemannian manifold (M, g) of neutral signature admitting a parallel degenerate plane field \(\mathfrak {D}\) of maximal dimension. Walker showed in [19] the existence of local coordinates \((x^1,x^2,x_{1'},x_{2'})\) so that \(\mathfrak {D} = {\text {span}}\{\partial _{x_{1'}},\partial _{x_{2'}}\}\) and the metric expresses as
The simplest examples of Walker manifolds are given by the so-called Riemannian extensions. We briefly review their construction as follows. Consider a surface \(\Sigma \) and let \(\pi :T^*\Sigma \rightarrow \Sigma \) be the projection from its cotangent bundle. Let \((p,\omega )\in T^*\Sigma \) denote a point in \(T^*\Sigma \), where \(p\in \Sigma \) and \(\omega \in T_p^*\Sigma \). For each vector field X on \(\Sigma \) the evaluation map is the function \(\iota X\in \mathcal {C}^\infty (T^*\Sigma )\) defined by \(\iota X(p,\omega ) = \omega (X_p)\). Two vector fields \(\bar{X}\) and \(\bar{Y}\) on \(T^*\Sigma \) satisfy \(\bar{X}=\bar{Y}\) if and only if they act on evaluation maps as \(\bar{X}(\iota Z)=\bar{Y}(\iota Z)\) for any vector field Z on \(\Sigma \). Given a vector field X on \(\Sigma \), its complete lift \(X^C\) is the vector field determined by the identity \(X^C(\iota Z) = \iota [X,Z]\). In the same way as vector fields on \(T^*\Sigma \) are characterized by their action on evaluation maps, (0, s)-tensor fields on \(T^*\Sigma \) are characterized by their action on complete lifts of vector fields. In particular, any (1, 1)-tensor field T on \(\Sigma \) induces a 1-form \(\iota T\) on \(T^*\Sigma \) characterized by \(\iota T (X^C) = \iota (TX)\) (see [20] for more details concerning this matter).
Riemannian extensions of torsion-free connections were introduced by Patterson and Walker in [18] as metrics on \(T^*\Sigma \) such that \(g_D(X^C,Y^C) = -\iota (D_X Y +D_Y X)\), where D is a torsion-free connection on the base manifold \(\Sigma \). Deformed Riemannian extensions are neutral signature metrics on \(T^*\Sigma \) such that \(g_{D,\Phi } = g_D +\pi ^*\Phi \), where \(\Phi \) is a symmetric (0, 2)-tensor field on the affine surface. Afifi showed in [1] that a Walker manifold with parallel null distribution \(\mathfrak {D}\) is locally isometric to a deformed Riemannian extension of an affine connection if and only if its curvature tensor satisfies \(R(\cdot ,\mathfrak {D})\mathfrak {D} = 0\). These metrics were further generalized in [8] as follows. Considering a symmetric (0, 2)-tensor field \(\Phi \) and (1, 1)-tensor fields T and S on an affine surface \((\Sigma ,D)\), the modified Riemannian extension is the neutral signature metric on \(T^*\Sigma \) defined by \(g_{D,\Phi ,T,S}=\iota T \circ \iota S + g_D +\pi ^*\Phi \), where ‘\(\circ \)’ denotes the symmetric product. Considering local coordinates \((x^1,x^2)\) on a neighbourhood \(\mathcal {U}\) in \(\Sigma \) and induced coordinates \((x^1,x^2,x_{1'},x_{2'})\) on \(\pi ^{-1}(\mathcal {U})\), one has
where \(T=T_i\!{}^j dx^i\otimes \partial _{x^j}\), \(S = S_i\!{}^jdx^i\otimes \partial _{x^j}\), \(\Phi = \Phi _{ij} dx^i\otimes dx^j\) and \({}^D\!\Gamma _{ij}\!{}^k\) are the Christoffel symbols of the affine connection D. Moreover, the Walker distribution is given by \(\mathfrak {D}={\text {ker}}\pi _*\). Furthermore, a Walker metric corresponds to the modified Riemannian extension of an affine connection if and only if \((\nabla _{\mathfrak {D}}R)(\mathfrak {D},\cdot )\mathfrak {D} = 0\).
2.1 Self-Dual Walker Manifolds
The existence of a parallel degenerate 2-dimensional distribution \(\mathfrak {D}\) on a neutral signature manifold (M, g) of dimension four naturally induces an orientation. We recall the discussion in [12]. Let \(p\in M\) and let \(\{u,v\}\) be an arbitrary basis of \(\mathfrak {D}_p\). Then the Hodge-star operator satisfies \(\star (u^*\wedge v^*)=\pm (u^*\wedge v^*)\), where \(u^*,v^*\in T_p^*M\) denote the corresponding dual forms. This way, any four-dimensional Walker manifold is naturally oriented by the self-duality of \(u^*\wedge v^*\). Let \((x^1,x^2,x_{1'},x_{2'})\) be local coordinates on a four-dimensional Walker manifold as in (1). Then the Walker orientation determined by \(\star (dx_{1'}\wedge dx_{2'}) = dx_{1'}\wedge dx_{2'}\) corresponds to the volume element \({\text {vol}}_g=dx^1\wedge dx^2\wedge dx_{1'}\wedge dx_{2'}\). Self-dual Walker manifolds were described in [8] as follows.
Theorem 1
([8, Theorem 7.1]) A four-dimensional Walker manifold is self-dual if and only if it is locally isometric to the cotangent bundle \(T^*\Sigma \) of an affine surface \((\Sigma ,D)\) with metric
where \(g_D\) denotes de Riemannian extension of the affine connection, X is a vector field on \(\Sigma \) and T and \(\Phi \) are a (1, 1)-tensor field and a symmetric (0, 2)-tensor field on \(\Sigma \), respectively.
Let \(\Sigma \) be a surface with local coordinates \((x^1,x^2)\) and consider \((x^1,x^2,x_{1'},x_{2'})\) the induced local coordinates on \(T^*\Sigma \). The canonical symplectic structure of the cotangent bundle determined by the tautological 1-form \(\theta = x_{k'}dx^k\) induces an orientation determined by the volume form \(d\theta \wedge d\theta = -dx^1\wedge dx^2\wedge dx_{1'}\wedge dx_{2'}\), which is the opposite of the orientation induced by the Walker structure given by \(\mathfrak {D}={\text {ker}}\pi _*\).
3 Bochner-Flat Para-Kähler Surfaces
Let (M, g, J) be a para-Kähler surface and denote \(\mathfrak {D}_\pm = {\text {ker}}(J\mp {\text {Id}})\). We consider Walker coordinates \((x^1,x^2,x_{1'},x_{2'})\) as in (1) and set the Walker distribution to be \(\mathfrak {D}=\mathfrak {D}_+\) so that \(J\vert _{\mathfrak {D}}={\text {Id}}\). We point out that para-Kähler surfaces are Walker manifolds but the converse is not true, since the parallelizability of \(\mathfrak {D} = \mathfrak {D}_+\) does not ensure the integrability of the complementary distribution \(\mathfrak {D}_-\). The almost para-Hermitian structures satisfying \(J\vert _{\mathfrak {D}}={\text {Id}}\) are locally parametrized by a real-valued function \(f(x^1,x^2,x_{1'},x_{2'})\) so that
Their associated Kähler 2-forms \(\Omega _f(X,Y)=g(J_fX,Y)\) are given by \(\Omega _f = (f-g_{12})dx^1\wedge dx^2 + dx_{1'}\wedge dx^1 + dx_{2'}\wedge dx^2\), thus
Therefore, \(d\Omega _f = 0\) if and only if \(f(x^1,x^2,x_{1'},x_{2'}) = g_{12}(x^1,x^2,x_{1'},x_{2'})+h(x^1,x^2)\) for some function \(h(x^1,x^2)\) and the almost paracomplex structure becomes
Considering an almost para-Hermitian structure given by (1) and (4), the associated Kähler 2-form is given by \(\Omega _h = h dx^1\wedge dx^2 + dx_{1'}\wedge dx^1 + dx_{2'}\wedge dx^2\). It is important to emphasize that the para-Kähler and Walker orientations are opposite. Indeed, the Kähler 2-form \(\Omega _h\) is anti-self-dual for the para-Kähler orientation determined by the paracomplex structure \(J_h\), but it is self-dual for the Walker orientation.
In order to describe Bochner-flat para-Kähler surfaces we consider the cotangent bundle \(T^*\Sigma \) of an affine surface \((\Sigma ,D)\) with metric \(g = \iota X(\iota {\text {Id}}\circ \iota {\text {Id}}) + \iota T \circ \iota {\text {Id}} + g_D + \pi ^*\Phi \) as in (2) and set the paracomplex structure satisfying the condition \(J\vert _{{\text {ker}}\pi _*} = {\text {Id}}\). The almost para-Hermitian structures defined by (1) and (4) are not para-Kähler in general. We use the notation \((\nabla _{\partial _{x^\alpha }}J_h)\partial _{x^\beta } = (\nabla J_h)_{\beta ;\alpha }{}^\gamma \partial _{x^\gamma }\) to denote the components of \(\nabla J_h\) on \(T^*\Sigma \) and \((D_{\partial _{x^i}}T)\partial _{x^j}=DT_{j;i}{}^k\partial _{x^k}\), \((D_{\partial _{x^i}}\Phi )(\partial _{x^j},\partial _{x^k})= D\Phi _{jk;i}\) to represent the covariant derivatives of the (1, 1)-tensor field T and the symmetric (0, 2)-tensor field \(\Phi \) on \(\Sigma \), respectively. Using the notation in Theorem 1, long but straightforward calculations show that:
Lemma 1
Let (M, g) be a self-dual Walker manifold of dimension four. Let \(J_h\) be an almost paracomplex structure given by \(J_h\vert _{{\text {ker}}\pi _*}={\text {Id}}\) so that \((g,J_h)\) is an almost para-Hermitian structure on M. Then the nonzero components of the covariant derivative \(\nabla J_h\) are given by
where \(X = X^i\partial _i\), \(T = T_j\!{}^i \partial _{x^i}\otimes dx^j\) and \(\Phi = \Phi _{ij} dx^i\otimes dx^j\) are the vector field, the (1, 1)-tensor field and the symmetric (0, 2)-tensor field on \(\Sigma \) given in Theorem 1, \(\mathcal {S}\) is the (1, 1)-tensor field on \(\Sigma \) defined as \(\mathcal {S}(Z):= D_Z X\), \(\hat{\Phi }(X,Y):=\Phi (TX, Y)\) and \({}^D\!\Gamma _{ij}\!{}^k\) are the Christoffel symbols of the affine connection.
Notice that the expressions in Lemma 1 are polynomials on the fiber coordinates \(x_{1'}\) and \(x_{2'}\) whose coefficients are functions of the base coordinates \(x^1\) and \(x^2\).
3.1 Bochner-Flat Para-Kähler Surfaces of Constant Scalar Curvature
It follows from Theorem 1 that the scalar curvature of a Bochner-flat para-Kähler surface is given by \(\tau = 12 \iota X + 3{{\,\mathrm{\mathrm tr}\,}}(T)\), where \(\iota X\) is the evaluation map of the vector field X. Therefore if a Bochner-flat para-Kähler surface has constant scalar curvature then the vector field X vanishes and T must have constant trace. If \(\tau \ne 0\) there exist local coordinates in which the (1, 1)-tensor field \(T = c {\text {Id}}\) with \(c\in \mathbb {R}\). In this situation, a Bochner-flat para-Kähler surface has constant paraholomorphic sectional curvature and so it is locally isometric to the cotangent bundle of a flat affine surface \((\Sigma ,D)\) endowed with a modified Riemannian extension \(g = c\iota {\text {Id}}\circ \iota {\text {Id}} + g_D\) (see [9, Theorem 2.2]).
Bochner-flat para-Kähler surfaces with \(\tau = 0\) are locally conformally flat. Working at a purely algebraic level, we consider \((V,\langle \cdot ,\cdot \rangle ,J)\) a para-Hermitian inner product space and a para-Kähler algebraic curvature tensor \(\mathcal {A}: V\times V \times V \times V \rightarrow \mathbb {R}\) so that \(\mathcal {A}(X,Y)\cdot J = J\cdot \mathcal {A}(X,Y)\). There are three non-flat locally conformally flat algebraic curvature models \((V,\langle \cdot ,\cdot \rangle , \mathcal {A})\) as follows.
\((\mathfrak {M})\): \(((V,\langle \,\cdot \,,\,\cdot \,\rangle , \mathcal {A})\) given by
with respect to pseudo-orthonormal a basis \(\{ u_1,u_2,u_3,u_4\}\) where the non-zero inner products are \(\langle u_1,u_2\rangle =1=-\langle u_3,u_4\rangle \).
(\(\mathfrak {N}_k\)): \(((V,\langle \,\cdot \,,\,\cdot \,\rangle , \mathcal {A})\) given by
with respect to an orthonormal basis \(\{ u_1,u_2,u_3,u_4\}\) where \(u_1,u_3\) are spacelike vectors and \(u_2,u_4\) are timelike vectors.
(\(\mathfrak {P}_k\)): \(((V,\langle \,\cdot \,,\,\cdot \,\rangle , \mathcal {A})\) given by
with respect to an orthonormal basis \(\{ u_1,u_2,u_3,u_4\}\) where \(u_1,u_3\) are spacelike vectors and \(u_2,u_4\) are timelike vectors.
It follows from Lemma 1 that if \(\tau = 0\), then the (1, 1)-tensor field T must be parallel and so the classification of Bochner-flat para-Kähler surfaces of constant scalar curvature is summarized as follows.
Theorem 2
([13, Theorem 4.2]) Let (M, g, J) be a Bochner-flat para-Kähler surface of constant scalar curvature. Then it is locally isometric to a Riemannian extension of the form \((T^*\Sigma ,g=\iota T \circ \iota {\text {Id}}+g_D)\) with paracomplex structure determined by \(J\vert _{{\text {ker}}\pi _*}={\text {Id}}\), where T is a parallel (1, 1)-tensor field on a flat affine surface \((\Sigma ,D)\). Moreover, one of the following holds:
-
(i)
\(T=0\) and (M, g, J) is flat.
-
(ii)
\(T = c{\text {Id}}\) and (M, g, J) has constant paraholomorphic sectional curvature \(H=c\).
-
(iii)
\(T^2=\kappa ^2{\text {Id}}\) and (M, g, J) is isometric to a product of two Lorentzian surfaces of constant opposite curvature, thus modelled on \((\mathfrak {P}_k)\).
-
(iv)
\(T^2 = 0\) and (M, g, J) is modelled on \((\mathfrak {M})\).
-
(v)
\(T^2=-\kappa ^2{\text {Id}}\) and (M, g, J) is modelled on \((\mathfrak {N}_k)\).
3.2 Some Examples of Bochner-Flat Para-Kähler Structures of Non-constant Scalar Curvature
Consider an affine surface \((\Sigma ,D)\) and let \((g,J_h)\) be an almost para-Hermitian structure on \(T^*\Sigma \) given by \(J_h\vert _{{\text {ker}}\pi _*} = {\text {Id}}\), where the metric \(g = \iota X (\iota {\text {Id}}\circ \iota {\text {Id}}) + \iota T\circ \iota {\text {Id}} + g_D + \pi ^*\Phi \) is given as in Theorem 1. Aimed to construct examples of Bochner-flat para-Kähler surfaces of non-constant scalar curvature we analyze the case where the (1, 1)-tensor field T is parallel. In this situation, the nonzero components of the covariant derivative of \(J_h\) reduce to
Since the tensor field T is parallel, it has constant trace and X must be nonzero so that \(\tau = 12 \iota X + 3{{\,\mathrm{\mathrm tr}\,}}(T)\) is non-constant.
If \(X^1 \ne 0\) it follows immediately from the expression of the coefficient of \(x_{1'}^2\) in \((\nabla J_h)_{1;2}{}^4\) above that \(\Phi _{22} = 0\). Knowing this, the expression of the coefficient of \(x_{1'}x_{2'}\) in \((\nabla J_h)_{1;2}{}^4\) shows that \(h=0\). The same coefficient in \((\nabla J_h)_{1;1}{}^4\) shows that \(\Phi _{11} = 0\) and now, focusing on the coefficient of \(x_{1'}^2\) in \((\nabla J_h)_{1;1}{}^4\) we see that \(\Phi _{12} = 0\). If \(X^2\ne 0\), proceeding analogously it follows that h and \(\Phi \) vanish identically.
Since both the function h and the symmetric (0, 2)-tensor field \(\Phi \) vanish, the components of the covariant derivative \(\nabla J_h\) reduce to
The linear terms in these two expressions show that the Ricci tensor of the affine surface must vanish identically. Therefore the affine connection is necessarily flat. Assume that T is trace-free. At this point, the components of \(\nabla J_h\) take the form
The existence of parallel (1, 1)-tensor fields on affine surfaces was studied in [9] showing that (besides the case where \(T=0\)) a trace-free parallel (1, 1)-tensor field on an affine surface \((\Sigma ,D)\) corresponds to one of the following.
-
(a)
An affine para-Kähler structure (\(\det (T) = -\kappa ^2<0\)), which in suitable adapted coordinates becomes \(T = \kappa (\partial _{x^1}\otimes dx^1-\partial _{x^2}\otimes dx^2)\).
-
(b)
An affine nilpotent Kähler structure (\(T^2=0\)), which in suitable adapted coordinates becomes \(T= \kappa \partial _{x^1}\otimes dx^2\).
-
(c)
An affine Kähler structure (\(\det (T)=\kappa ^2>0\)), which in suitable adapted coordinates becomes \(T=\kappa (\partial _{x^2}\otimes dx^1 - \partial _{x^1}\otimes dx^2)\).
Straightforward calculations now show that, for any case described above, there exist local coordinates in which the (1, 1)-tensor filed \(\mathcal {S} = D X\) takes the form \(\mathcal {S} = \lambda {\text {Id}}\) for some function \(\lambda \in \mathcal {C}^\infty (\Sigma )\) and the scalar curvature is given by \(\tau = \iota X\).
We summarize the discussion above in the following
Theorem 3
Let \((\Sigma ,D)\) be an affine surface and let \((g,J_h)\) be an almost para-Hermitian structure on \(T^*\Sigma \) such that
If T is parallel and \({{\,\mathrm{\mathrm tr}\,}}(T)=0\) then \((T^*\Sigma ,g,J_h)\) is a Bochner-flat para-Kähler surface if and only if \(h= 0\), \(\Phi =0\), the affine connection D is flat and \(\mathcal {S} = \lambda {\text {Id}}\) for some \(\lambda \in \mathcal {C}^\infty (\Sigma )\), being \(\mathcal {S}(Z) = D_Z X\).
Remark 1
If the affine connection D is flat, there exist local coordinates on \(\Sigma \) so that all the Christoffel symbols are zero. After a suitable linear transformation on the coordinates one can set T being of one of the forms described above. Straightforward calculations show that there exist suitable adapted coordinates in which Bochner-flat para-Kähler structures determined by a trace-free parallel (1, 1)-tensor field T are given by the vector field \(X = a x^i\partial _{x^i}\), where \(a\in \mathbb {R}\) and thus \(\mathcal {S} = a {\text {Id}}\). We subsequently examine the different possibilities.
-
(a)
Let T be an affine para-Kähler structure on a flat affine surface \((\Sigma ,D)\) and take local coordinates so that \({}^D\!\Gamma _{ij}\!{}^k=0\), \(T = \kappa (\partial _{x^1}\otimes dx^1-\partial _{x^2}\otimes dx^2)\) and \(X = a x^i\partial _{x^i}\). Then the Bochner-flat metric induced on \(T^*\Sigma \) is given by
$$ \begin{array}{l} g = x_{1'}^2(ax_{1'}x^1 \!+\! ax_{2'}x^2 \!+\! \kappa ) dx^1\!\otimes dx^1 \!+\! x_{2'}^2(ax_{1'}x^1 \!+\! ax_{2'}x^2-\kappa )dx^2\!\otimes dx^2 \\ [.05in] \phantom {g=} \!+\! ax_{1'}x_{2'}(x_{1'}x^1 \!+\! x_{2'}x^2)(dx^1\!\otimes dx^2 \!+\! dx^2\!\otimes dx^1) \!+\! dx^1\!\otimes \! dx_{1'} \!+\! dx^2\!\otimes \! dx_{2'}. \end{array} $$A straightforward calculation shows that the Ricci curvatures are given by
$$ \lambda _\pm = 2\iota X \pm \sqrt{(\iota X)^2 + 2\iota (TX) + \kappa ^2}, $$and the Ricci operator diagonalizes with real eigenvalues on the zero section of \(T^*\Sigma \). The curvature on the zero section of the cotangent bundle corresponds to that of a locally conformally flat para-Kähler surface determined by an affine para-Kähler structure, thus it is modelled on \((\mathfrak {P}_k)\).
-
(b)
Let T be an affine nilpotent Kähler structure on a flat affine surface \((\Sigma ,D)\) and take local coordinates so that \({}^D\!\Gamma _{ij}\!{}^k=0\), \(T= \kappa \partial _{x^1}\otimes dx^2\) and \(X = a x^i\partial _{x^i}\). Then the Bochner-flat metric induced on \(T^*\Sigma \) is given by
$$ \begin{array}{l} g = ax_{1'}^2(x_{1'}x^1\!+\! x_{2'}x^2) dx^1\otimes dx^1 \!+\! x_{2'}(ax_{2'}(x_{1'}x^1\!+\!x_{2'}x^2) + \kappa x_{1'})dx^2\otimes dx^2 \\ \phantom {g=} \!+\!\tfrac{1}{2}x_{1'}(2ax_{2'}(x_{1'}x^1 \!+\! x_{2'}x^2)\!+\! \kappa x_{1'})(dx^1\otimes dx^2 \!+\! dx^2\otimes dx^1) \\ \phantom {g=}\!+\! dx^1\otimes dx_{1'} \!+\! dx^2\otimes dx_{2'}. \end{array} $$A straightforward calculation shows that the Ricci curvatures are given by
$$ \lambda _\pm = 2\iota X \pm \sqrt{(\iota X)^2 + 2\iota (TX) }, $$and the Ricci operator is two-step nilpotent on the zero section of \(T^*\Sigma \). The curvature on the zero section of the cotangent bundle corresponds to that of a locally conformally flat para-Kähler surface determined by an affine nilpotent Kähler structure, thus it is modelled on \((\mathfrak {M})\).
-
(c)
Let T be an affine Kähler structure on a flat affine surface \((\Sigma ,D)\) and take local coordinates so that \({}^D\!\Gamma _{ij}\!{}^k=0\), \(T=\kappa (\partial _{x^2}\otimes dx^1 - \partial _{x^1}\otimes dx^2)\) and \(X = a x^i\partial _{x^i}\). Then the Bochner-flat metric induced on \(T^*\Sigma \) is given by
$$ \begin{array}{l} g = x_{1'}(a x_{1'}(x_{1'}x^1+x_{2'}x^2)+\kappa x_{2'}) dx^1\otimes dx^1 \\ \phantom {g=} \!+\! x_{2'}(ax_{2'}(x_{1'}x^1+x_{2'}x^2)-\kappa x_{1'})dx^2\otimes dx^2 \\ \phantom {g=} \!+\! \left( ax_{1'}x_{2'}(x_{1'}x^1 \!+\! x_{2'}x^2)\!+\!\tfrac{1}{2}(x_{2'}^2\!-\! x_{1'}^2)\kappa \right) (dx^1\otimes dx^2 \!+\! dx^2\otimes dx^1) \\ \phantom {g=}\!+\! dx^1\otimes dx_{1'} \!+\! dx^2\otimes dx_{2'}. \end{array} $$A straightforward calculation shows that the Ricci curvatures are given by
$$ \lambda _\pm = 2\iota X \pm \sqrt{(\iota X)^2 + 2\iota (TX) - \kappa ^2}, $$and the Ricci operator has complex eigenvalues on the zero section of \(T^*\Sigma \). The curvature on the zero section of the cotangent bundle corresponds to that of a locally conformally flat para-Kähler surface determined by an affine Kähler structure, thus it is modelled on \((\mathfrak {N}_k)\).
Remark 2
In the case where T is a parallel (1, 1)-tensor field with nonzero trace on \((\Sigma ,D)\), we denote \(T^0 = T - \frac{{{\,\mathrm{\mathrm tr}\,}}(T)}{2}{\text {Id}}\) the traceless part of T so that \(T = T^0 + \mu {\text {Id}}\), being \({{\,\mathrm{\mathrm tr}\,}}(T) = 2\mu \). Since the affine connection D is flat, there exist local coordinates in which all the Christoffel symbols \({}^D\!\Gamma _{ij}\!{}^k\) are zero. After a suitable linear transformation on the coordinates we can set \(T^0\) being of one of the forms described in (a), (b) and (c) above. Straightforward calculations show that there exist suitable adapted coordinates in which Bochner-flat para-Kähler structures determined by a parallel (1, 1)-tensor field T with \({{\,\mathrm{\mathrm tr}\,}}(T)\ne 0\) correspond to one of the following situations.
-
(a)
If \(T^0\) is an affine para-Kähler structure then the (1, 1)-tensor fields T and \(\mathcal {S}\) commute and there exist local coordinates in which \(T = (\mu + \kappa )\partial _{x^1}\otimes dx^1 + (\mu -\kappa )\partial _{x^2}\otimes dx^2\) and \(X = ax^1 \partial _{x^1} + (a + \frac{\kappa \mu }{2})\partial _{x^2}\). The Bochner-flat metric induced on \(T^*\Sigma \) has Ricci curvatures given by
$$ \lambda _{\pm } = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX) + \kappa ^2}, $$where \(K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}\). The Ricci operator diagonalizes with real eigenvalues \(\lambda _\pm = \frac{3}{2}\mu \pm \kappa \) on the zero section of the cotangent bundle \(T^*\Sigma \).
-
(b)
If \(T^0\) is an affine nilpotent Kähler structure then the (1, 1)-tensor fields T and \(\mathcal {S}\) commute and there exist local coordinates in which \(T = \mu (\partial _{x^1}\otimes dx^1 + \partial _{x^2}\otimes dx^2) + \kappa \partial _{x^1}\otimes dx^2\) and \(X = \left( a x^1 -\frac{\kappa \mu }{4}x^2\right) \partial _{x^1} + ax^2\partial _{x^2}\). The Bochner-flat metric induced on \(T^*\Sigma \) has Ricci curvatures given by
$$ \lambda _\pm = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX)}, $$where \(K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}\). On the zero section of the cotangent bundle the Ricci operator has a single eigenvalue \(\lambda = \frac{3}{2}\mu \) that is a double root of its minimal polynomial.
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(c)
If \(T^0\) is an affine Kähler structure then the (1, 1)-tensor fields T and \(\mathcal {S}\) commute and there exist local coordinates in which \(T = \mu (\partial _{x^1}\otimes dx^1 + \partial _{x^2}\otimes dx^2) + \kappa (\partial _{x^2}\otimes dx^1 - \partial _{x^1}\otimes dx^2)\) and \(X = \left( a x^1 + \frac{\kappa \mu }{4}x^2\right) \partial _{x^1} + \left( a x^2 - \frac{\kappa \mu }{4}x^1\right) \partial _{x^2}\). The Bochner-flat metric induced on \(T^*\Sigma \) has Ricci curvatures given by
$$ \lambda _\pm = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX)-\kappa ^2}, $$where \(K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}\). The Ricci operator is complex-diagonalizable with eigenvalues \(\lambda _\pm = \frac{3}{2}\mu + \sqrt{-\kappa ^2}\) on the zero section of \(T^*\Sigma \).
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(d)
If \(T^0 = 0\) there exist local coordinates in which the (1, 1)-tensor field T is a multiple of the identity \(T = \mu {\text {Id}}\) and \(X = ax^i\partial _{x^i}\). The Bochner-flat metric induced on \(T^*\Sigma \) has Ricci curvatures given by
$$ \lambda _\pm = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX)}, $$where \(K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}\). On the zero section of the cotangent bundle, the Ricci operator is diagonalizable with a single real eigenvalue \(\lambda = \frac{3}{2}\mu \). Therefore the paraholomorphic sectional curvature is constant on the zero section of \(T^*\Sigma \).
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Acknowledgements
This work has been supported by projects PID2019-105138GB-C21(AEI/FEDER, Spain), and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain).
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Ferreiro-Subrido, M. (2022). Bochner-Flat Para-Kähler Surfaces. In: Albujer, A.L., Caballero, M., García-Parrado, A., Herrera, J., Rubio, R. (eds) Developments in Lorentzian Geometry. GELOMA 2021. Springer Proceedings in Mathematics & Statistics, vol 389. Springer, Cham. https://doi.org/10.1007/978-3-031-05379-5_5
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